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Concept
Measurable space
Summary
In mathematics, a measurable space or Borel space is a basic object in measure theory. It consists of a set and a σ-algebra, which defines the subsets that will be measured.
Consider a set and a σ-algebra on Then the tuple is called a measurable space.
Note that in contrast to a measure space, no measure is needed for a measurable space.
Look at the set:
One possible -algebra would be:
Then is a measurable space. Another possible -algebra would be the power set on :
With this, a second measurable space on the set is given by
If is finite or countably infinite, the -algebra is most often the power set on so This leads to the measurable space
If is a topological space, the -algebra is most commonly the Borel -algebra so This leads to the measurable space that is common for all topological spaces such as the real numbers
The term Borel space is used for different types of measurable spaces.
Official source
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